Fresnel Sine Integral Function of Zero

Theorem

$\map {\operatorname S} 0 = 0$

where $\operatorname S$ denotes the Fresnel sine integral function.

Proof

By Fresnel Sine Integral Function is Odd, $\operatorname S$ is an odd function.

Therefore, by Odd Function of Zero is Zero:

$\map {\operatorname S} 0 = 0$

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Fresnel Sine Integral $\ds \map {\operatorname S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$: $35.20$